What is the error for the estimate of the probability of a binomial distribution?

Question

I want to estimate the probability $p$ of a Binomial distribution $B(n,p)$. I draw $n$ samples and get $s$ successes. The estimate for $p$ is $$\hat{p} = s/n.$$ What is the variance of $\hat{p}$?

Answer 1

In your definition, $s$ is a random variable, whose mean is $np$ and whose variance is $np(1-p)$ (this follows from the fact that the number of successes is a binomial $\mathcal{B}(n,p)$ random variable).

Thus,

\begin{align} \text{var}(\hat{p})&=\text{var}\left(\frac{s}{n}\right)\\\ &=\frac{1}{n^{2}}\text{var}(s)\\ &=\frac{np(1-p)}{n^{2}}\\ &=\frac{p(1-p)}{n}. \end{align}

Answer 2

When the sample proportion $\hat p$ is used to estimate $\pi$, the standard error is $$= \sqrt{\dfrac{\pi (1-\pi)}{n}}$$

Standard error from data

Unfortunately, the standard error of $\hat p$ involves $\pi$, and this is unknown in practical problems. To get a numerical value for the standard error, we must therefore replace $\pi$ with our best estimate of its value, $\hat p$.

standard error is the calculated using the standard deviation (square root of variance that you asked for)

$$= \sqrt{\dfrac{\hat p (1-\hat p)}{n}}$$

Good Luck